A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Bjorner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.

Herzog and Takayama constructed an explicit resolution for the ideals with a regular linear quotient. These ideals include all matroidal and stable ideals. The resolutions of matroidal and stable ideals are known to be cellular. In this note, we show that the Herzog–Takayama resolution is also cellular.

The clique vector c(G) of a graph G is the sequence (c(1), c(2), ..., c(d)) in N-d, where c(i) is the number of cliques in G with i vertices and d is the largest cardinality of a clique in G. In this note, we use tools from commutative algebra to characterize all possible clique vectors of k-connected chordal graphs.

In this paper we verify a conjecture by Kozlov [D.N. Kozlov, Convex Hulls of f- and beta-vectors, Discrete Comput. Geom. 18 (1997) 421-431], which describes the convex hull of the set of face vectors of r-colorable complexes on n vertices. As part of the proof we derive a generalization of Turn's graph theorem.

Let k be a field and let A be a standard N-graded k-algebra. Using numerical information of some invariants in the primary decomposition of 0 in A, namely the so-called dimension filtration, we associate a bivariate polynomial BW(A;t,w), that we call the Björner-Wachs polynomial, to A.It is shown that the Björner-Wachs polynomial is an algebraic counterpart to the combinatorially defined h-triangle of finite simplicial complexes introduced by Björner & Wachs. We provide a characterisation of sequentially Cohen-Macaulay algebras in terms of the effect of the reverse lexicographic generic initial ideal on the Björner-Wachs polynomial. More precisely, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the reverse lexicographic generic initial ideal. We conclude by discussing some connections with the Hilbert series of local cohomology modules, extremal Betti numbers and combinatorial Alexander duality.

Convex hull of face vectors of colored complexes. In this paper we verify a conjecture by Kozlov (Discrete ComputGeom18(1997) 421–431), which describes the convex hull of theset of face vectors ofr-colorable complexes onnvertices. As partof the proof we derive a generalization of Turán’s graph theorem.

Cellular structure for the Herzog–Takayama Resolution. Herzog and Takayama constructed explicit resolution for the ide-als in the class of so called ideals with a regular linear quotient.This class contains all matroidal and stable ideals. The resolu-tions of matroidal and stable ideals are known to be cellular. Inthis note we show that the Herzog–Takayama resolution is alsocellular.

Clique Vectors ofk-Connected Chordal Graphs. The clique vectorc(G)of a graphGis the sequence(c1,c2,...,cd)inNd, whereciis the number of cliques inGwithivertices anddis the largest cardinality of a clique inG. In this note, we usetools from commutative algebra to characterize all possible cliquevectors ofk-connected chordal graphs.

To every squarefree monomial ideal one can associate a hypergraph. In this paper we show that the Hilbert series of a squarefree monomial ideal can be obtained from the so-called edge induced polynomial of the associated hypergraph.

This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.